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Chemical and Process Engineering Research www.iiste.orgISSN 2224-7467 (Paper) ISSN 2225-0913 (Online)Vol 2, 2012 Effect of Radiation and Chemical Reaction on Transient MHD Free Convective Flow over a Vertical Plate Through Porous Media N.Sandeep A.Vijaya Bhaskar Reddy V.Sugunamma* S.V.U College of Mathematical and Physical Sciences, Tirupati, A.P., India. Email: nsreddy.dr@gmail.comAbstract:This paper analyze the Magneto hydrodynamic, Radiation and chemical reaction effects on unsteady flow,heat and mass transfer characteristics in a viscous, in compressible and electrically conduction fluid over asemi-infinite vertical porous plate through porous media. The porous plate is subjected to a transversevariable suction velocity. The transient, non-linear and coupled governing equations have been solvedadopting a perturbative series expansion about a small parameter, ε. The effects of governing parameters onthe flow variables are discussed graphically.Keywords: Transient velocity, MHD, Chemicalreaction, Radiation and Rarefaction Parameter.Introduction: Natural convection flow over vertical surfaces immersed in porous media has paramount importancebecause of its potential applications in soil physics, geohydrology, and filtration of solids from liquids,chemical engineering and biological systems. Study of fluid flow in porous medium is based upon theempirically determined Darcy’s law. Such flows are considered to be useful in diminishing the freeconvection, which would otherwise occur intensely on a vertical heated surface. In addition, recentdevelopments in modern technology have intensified more interest of many researchers in studies of heat andmass transfer in fluids due to its wide applications in geothermal and oil reservoir engineering as well as othergeophysical and astrophysical studies. Cramer, K. R. and Pai, S. I. [1] taken transverse applied magnetic field and magnetic Reynoldsnumber are assumed to be very small, so that the induced magnetic field is negligible Muthucumaraswamy etal. [2] have studied the effect of homogenous chemical reaction of first order and free convection on theoscillating infinite vertical plate with variable temperature and mass diffusion. Sharma [3] investigate theeffect of periodic heat and mass transfer on the unsteady free convection flow past a vertical flat plate in slip-flow regime when suction velocity oscillates in time. Chaudhary and Jha [4] studied the effects of chemicalreactions on MHD micropolar fluid flow past a vertical plate in slip-flow regime. Anjalidevi et al. [5] haveexamined the effect of chemical reaction on the flow in the presence of heat transfer and magnetic field.Muthucumaraswamy et al. [6] have investigated the effect of thermal radiation effects on flow past animpulsively started infinite isothermal vertical plate in the presence of first order chemical reaction. Moreover, Al-Odat and Al-Azab [7] studied the influence of magnetic field on unsteady freeconvective heat and mass transfer flow along an impulsively started semi-infinite vertical plate taking intoaccount a homogeneous chemical reaction of first order.. The effect of radiation on the heat and fluid flowover an unsteady stretching surface has been analyzed by El-Aziz [8]. Singh et. al. [9] studied the heattransfer over stretching surface in porous media with transverse magnetic field. Singh et. al. [10] and [11] alsoinvestigated MHD oblique stagnation-point flow towards a stretching sheet with heat transfer for steady andunsteady cases. Elbashbeshy et. al. [12] investigated the effects of thermal radiation and magnetic field onunsteady boundary layer mixed convection flow and heat transfer problem from a vertical porous stretchingsurface. Ahmed Sahin studied influence of chemical reaction on transient MHD free Convective flow over avertical plate. Recently, The chemical reaction, heat and mass transfer on MHD flow over a verticalstretching surface with heat source and thermal stratification have been presented by Kandasamy et al.[13].The opposing buoyancy effects on simultaneous heat and mass transfer by natural convection in a fluidsaturated porous medium investigated by Angirasa et al.[14]. Ahmed [15] investigates the effects of unsteadyfree convective MHD flow through a porous medium bounded by an infinite vertical porous plate. AhmedSahin [16] studied the Magneto hydrodynamic and chemical reaction effects on unsteady flow, heat and masstransfer characteristics in a viscous, incompressible and electrically conduction fluid over a semi-infinitevertical porous plate in a slip-flow regime. 1
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Chemical and Process Engineering Research www.iiste.orgISSN 2224-7467 (Paper) ISSN 2225-0913 (Online)Vol 2, 2012 The objective of the present study is to investigate the effect of various parameters like chemicalReaction parameter, thermal Grashof number, mass Grashof number, rarefaction parameter, magnetic fieldparameter, radiation parameter, suction parameter on convective heat transfer along an inclined plateembedded in porous medium. The governing non-linear partial differential equations are first transformedinto a dimensionless form and thus resulting non-similar set of equations has been solved using theperturbation technique. Results are presented graphically and discussed quantitatively for parameter values ofpractical interest from physical point of view.Mathematical Analysis: Consider the unsteady two dimensional MHD free convective flow of a viscous incompressible,electrically conducting and radiating fluid in an optically thin environment past an infinite heated verticalporous plate embedded in a porous mediumin presence of thermal and concentration buoyancy effects . Let the x-axis be taken in vertically upwarddirection along the plate and y-axis is normal to the plate. It is assumed that there exist a homogeneouschemical reaction of first order with constant rate K r between the diffusing species and the fluid .A uniformmagnetic field is applied in the direction perpendicular to the plate. The viscous dissipation and the Jouleheating effects are assumed to be negligible in the energy equation. The transverse applied magnetic field andmagnetic Reynolds number are assumed to be very small, so that the induced magnetic field is negligible [22]. Also it is assumed that there is no applied voltage, so that the electric field is absent. The concentration ofthe diffusing species in the binary mixture is assumed to be very small in comparison with the other chemicalspecies, which are present, and hence the Soret and Dufour effects are negligible and the temperature in thefluid flowing is governed by the energy concentration equation involving radiative heat temperature. Underthe above assumptions as well as Boussinesq’s approximation, the equations of conservation of mass,momentum, energy and concentration governing the free convection boundary layer flow over a verticalporous plate in porous medium can be expressed as:v1 0 (1)y1u1 v 1 Aeit u g T 1 T 1 g C C u 1 2 1 o2 1 v 1 o 1 1 1 u 1u (2)t1 y1 y 2 k T 1 T 1 2T 1 q1C p v0 1 Aeit 1 k r (3) t1 y y2 y1C1 C1 2C 1 v0 1 Aeit 1 D K 1 C1 C 1 (4)t1 y y2Where g , T , C , B0 , D, , k , C p , , , c , q and Kr are acceleration due to gravity ,fluid temperature,species concentration, magnetic field, chemical molecular diffusivity, electrical conductivity, thermalconductivity, specific heat constant pressure, kinematic viscosity, density, coefficient of volume expansionfor heat transfer, volumetric coefficient of expansion with species concentration, radiative heat flux andchemical reaction parameter respectively.The corresponding boundary conditions of the problem are u1 u1 L1 1 , T 1 Tw Tw T ei t 1 1 1 y C Cw Cw C eit at y1 0 1 1 1 1u1 0, T 1 T , C1 C at y1 1 1 (5) 2
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Chemical and Process Engineering Research www.iiste.orgISSN 2224-7467 (Paper) ISSN 2225-0913 (Online)Vol 2, 2012 1 1 1 1Where Tw andT is the temperature at the wall and infinity, Cw andC is the species concentration at thewall and at infinity respectively.By using Rosseland approximation the radiative heat flux q1 is given by r 4 s Tw4q Where s is the Stefen Boltzmann constant and ke is the mean absorption coefficient. 1 3ke y rBy expanding Tw4 in to the Taylor series about T which after neglecting higher order terms takes the 1form Tw 4 4T Tw 3T . 3 4From the equation of continuity (1), it is clear that the suction velocity at the plate is either a constant or afunction of time only. Hence, the suction velocity normal to the plate is assumed to be in the formv1 v0 1 Aeit (6)We now introduce the following non-dimensional quantities into the equations (1) to (5) v0 y1 u1 C py ,u , t t 1v0 4 , 41 v0 , , Pr 2 2 , v0 k T 1 T 1 C 1 C 1 g Tw T 1 1 g 1 Cw C 1 1 1 , 1 , Gr , Gc , Tw T1 Cw C1 3 v0 3 v0 02 v L1 k1 k v 2 kke v02M , Sc , h 0 , K r 2 , 20 , R . (7) v02 D v0 4 sT 3 2The governing equations (2) to (4) can be rewritten in the non-dimensional form as follows1 u u 2u 1 Aeit Gr Gc ( M )u (8)4 t y y 21 it 1 4 2 1 Ae 1 (9)4 t y Pr 3R y 21 1 2 1 Aeit K r (10)4 t y Sc y 2The transformed boundary conditions are uuh 1 eit y, 1 eit at y 0u 0, 0, 0 at y (11)Solution of the Problem: The equations (8) to (10) are coupled, non-linear partial differential equations and these cannot be solved inclosed form. However, these equations can be reduced to a set of ordinary differential equations, which canbe solved analytically. So this can be done, when the amplitude of oscillations ( <<1) is very small, we canassume the solutions of flow velocity u, temperature field and concentration in the neighborhood of theplate as:f y, t f 0 y eit f1 y ......, (12)Where f stands for u, or 3
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Chemical and Process Engineering Research www.iiste.orgISSN 2224-7467 (Paper) ISSN 2225-0913 (Online)Vol 2, 2012 4 10 Pr 1 11 0 0 (13) 3R 4 4 4 1111 Pr 1 11 i Pr 1 1 4 2 A Pr 1 0 (14) 3R 3R 3R u0 u0 (M )u0 Gr0 Gc0 11 1 (15)u1 u1 (M ) i 4 u1 Gr1 11 1 (16)0 Sc0 Kr Sc0 0 11 1 (17)1 Sc1 Sc Kr i 4 1 ASc0 11 1 1 (18)where prime denotes differentiation with respect to y. The corresponding boundary conditions are u u u0 h 0 , u1 h 1 ,0 1 y y 1 1, 0 1, 1 1 at y 0u0 0, u1 0,0 0,1 0, 0 0, 1 0 at y (19)The solutions of the equations (13) to (18) under the boundary the conditions (19) are 4 Pr 1 y0 y e 3R , (20)0 y e ny , (21) 4 Pr 1 y1 y B1e y B2e 3R , (22) 4 Pr 1 yu0 y B5e qy B3e 3R B4e ny , (22)1 y B7 e y B6e ny , (23) 4 Pr 1 yu1 y B15e y B10e y B11e y B12e qy B13e 3R B14e ny (24)Results and Discussion: The effects of governing parameters like magneticfield, chemical reaction, rarefaction parameter,suction parameter, thermal Grashof number as well as mass Grashof number porosity parameter and radiationparameter on the transient velocity have been presented in the respective Figures 1 to 20 for both the cases of air and water and in presence of foreign species Sc = 0.22 , t . Here we restricted our discussion to 2the aiding of favourable case only, for fluids with Prandtl number Pr = 0.71 which represent air at 20 0 C at 1atmosphere and for fluids Pr = 7 which represent water. The value of thermal Grashof number Gr is taken tobe positive, which corresponds to the cooling of the plate. The diffusing chemical species of most commoninterest in air has Schmidt number (Sc) and is taken for Hydrogen (Sc = 0.22), Oxygen (Sc = 0.66), andCarbon dioxide (Sc = 0.94). Figures (1) and (2) shows the effect of Grashof number on transient velocity for Pr = 0.71 and Pr =7.Here Chemical reaction parameter (Kr=0.1), Magnetic field parameter (M=1),Radiation parameter(R=3),Suction parameter (A=0.5),Refraction parameter(h = 0.2),Porosity parameter ( =0.1),Frequency 4
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Chemical and Process Engineering Research www.iiste.orgISSN 2224-7467 (Paper) ISSN 2225-0913 (Online)Vol 2, 2012parameter ( =30),Mass Grashof number (Gc=5) on the velocity profiles . We observe that the velocitygradient at the surface increase with the increase of Grashof number. Figures (3) and (4) represents the effect of porosity parameter on transient velocity for Pr =0.71 and Pr =7 .Here Chemical reaction parameter (Kr=0.1), Magnetic field parameter (M=1), Radiationparameter (R=2), Suction parameter (A=0.5), Refraction parameter(h = 0.2), Frequency parameter( =30),Mass Grashof number (Gc=5),Thermal Grashof number (Gr=5) on the velocity profiles . Weobserve that the velocity gradient at the surface increase with the decrease of porosity Parameter. Figures (5) and (6) shows the effect of Radiation parameter on transient velocity for Pr = 0.71and Pr =7 .Here Chemical reaction parameter (Kr=0.1), Magnetic field parameter (M=1), Suction parameter(A=0.5), Refraction parameter (h = 0.2), Porosity parameter ( =0.1), Frequency parameter ( =30), MassGrashof number (Gc=5) on the velocity profiles. We observe that the velocity gradient at the surface increasewith the decrease of radiation parameter but it is reversed in case of water. Figures (7) and (8) represents the effect of Chemical reaction parameter on transientvelocity for Pr = 0.71 and Pr =7. Magnetic field parameter (M=1), Radiation parameter (R=2), Suctionparameter (A=0.5), Refraction parameter (h = 0.2), Frequency parameter ( =30),Mass Grashof number(Gc=5),Thermal Grashof number (Gr=5) on the velocity profiles . We observe that the velocity high at thesurface with the decrease of Chemical reaction parameter. Figures (9) and (10) shows the effect of Magnetic field parameter on transient velocity for Pr = 0.71and Pr =7 .Here Chemical reaction parameter (Kr=0.1), Radiation parameter (R=2), Suction parameter(A=0.5),Refraction parameter(h = 0.2),Porosity parameter ( =0.1),Frequency parameter ( =30),MassGrashof number (Gc=5) on the velocity profiles . We observe that the velocity gradient at the surfaceincreases with the decrease of Magnetic parameter. Figures (11) and (12) shows the effect of rarefaction parameter on transient velocity for Pr = 0.71 andPr =7 .Here Chemical reaction parameter (Kr=0.1), Magnetic field parameter (M=1),Radiation parameter(R=3),Suction parameter (A=0.5),Refraction parameter(h = 0.2),Porosity parameter ( =0.1),Frequencyparameter ( =30),Mass Grashof number (Gc=5) on the velocity profiles . We observe that the velocitygradient at the surface decrease with the increase of rarefaction parameter. From figure (13) it is observed thateffect of suction parameter on velocity .Here velocity profiles decreases decreases in water and flow istogether. From figures (14) and (15) we observe the same results as in figures (1) and (2). Figure (16) shows the effect of porosity parameter on temperature profiles for Pr = 0.71 and Pr =7.Here Chemical reaction parameter (Kr=0.1), Magnetic field parameter (M=1),Radiation parameter(R=3),Suction parameter (A=0.5),Refraction parameter(h = 0.2),Porosity parameter ( =0.1),Frequencyparameter ( =30),Mass Grashof number (Gc=5) on the velocity profiles . Figure (17) shows the dimensionless concentration profiles (φ) for chemical reaction (K), Schmidtnumber and suction parameter (A) against frequency parameter. A decrease in concentration with increasingK as well as Sc is observed from this figure, while the concentration rises with the increase of suctionparameter. This is because as the suction rate is increased, more warm fluid is taken away from the boundarylayer. Also, it is noted that the concentration boundary layer becomes thin as the Schmidt number as well aschemical reaction parameter increases. 5
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Chemical and Process Engineering Research www.iiste.orgISSN 2224-7467 (Paper) ISSN 2225-0913 (Online)Vol 2, 2012 10 2.5 Gr=5,Pr=0.71 K=0.2,Pr=7 Gr=10,Pr=0.71 K=0.4,Pr=7 8 K=0.6,Pr=7 Gr=15,Pr=0.71 2 Gr=20,Pr=0.71 K=0.8,Pr=7 6 1.5 4 u u 2 1 0 0.5 -2 0 -4 0 2 4 6 8 10 12 0 2 4 6 8 10 12 y y Fig (4): Effect of porosity parameter on Fig (1): Effect of Grashof number on transient velocity for Pr=7 transient velocity for Pr=0.71. 3.5 2.5 R=1,Pr=0.71 Gr=5,Pr=7.0 3 R=1.5,Pr=0.71 Gr=10,Pr=7.0 R=2,Pr=0.71 Gr=15,Pr=7.0 2.5 R=3,Pr=0.71 2 Gr=20,Pr=7.0 2 1.5 1.5 u 1 u 1 0.5 0 0.5 -0.5 -1 0 2 4 6 8 10 12 y 0 0 2 4 6 8 10 12 y Fig (5): Effect of radiation parameter on Fig (2): Effect of Grashof number on transient velocity for Pr=0.71. transient velocity for Pr=71. 1.6 R=1,Pr=7 4 R=1.5,Pr=7 1.4 K=0.2,Pr=0.71 R=2,Pr=7 3.5 K=0.4,Pr=0.71 R=3,Pr=7 K=0.6,Pr=0.71 1.2 3 K=0.8,Pr=0.71 1 2.5 0.8 u 2 u 0.6 1.5 1 0.4 0.5 0.2 0 0 0 2 4 6 8 10 12 -0.5 y 0 2 4 6 8 10 12 y Fig (6): Effect of radiation parameter onFig (3): Effect of porosity parameter on transient transient velocity for Pr=7velocity for Pr=0.71. 6
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Chemical and Process Engineering Research www.iiste.orgISSN 2224-7467 (Paper) ISSN 2225-0913 (Online)Vol 2, 2012 2.5 2.5 Kr=0,Pr=7 M=0,Pr=7 Kr=0.1,Pr=7 M=1,Pr=7 Kr=0.5,Pr=7 M=3,Pr=7 2 2 Kr=1,Pr=7 M=5,Pr=7 1.5 1.5 u u 1 1 0.5 0.5 0 0 0 2 4 6 8 10 12 0 2 4 6 8 10 12 y y Fig (7): Effect of Chemical Reaction Fig (10): Effect of magnetic parameter parameter on transient velocity for Pr=7 on transient velocity for Pr=7 4 4 Kr=0,Pr=0.71 h=0,Pr=0.71 3.5 Kr=0.1,Pr=0.71 2 h=0.5,Pr=0.71 Kr=0.5,Pr=0.71 h=1,Pr=0.71 Kr=1,Pr=0.71 0 h=3,Pr=0.71 3 -2 2.5 -4 2 u u -6 1.5 -8 1 -10 0.5 -12 0 -14 0 2 4 6 8 10 12 0 2 4 6 8 10 12 y y Fig (8): Effect of Chemical Reaction Fig (11): Effect of rarefaction parameter parameter on transient velocity for on transient velocity for Pr=0.71 Pr=0.71 2.5 4 h=0,Pr=7 M=0,Pr=0.71 h=0.5,Pr=7 M=1,Pr=0.71 2 h=1,Pr=7 3.5 M=3,Pr=0.71 h=3,Pr=7 M=5,Pr=0.71 3 1.5 2.5 1 u 2 u 0.5 1.5 1 0 0.5 -0.5 0 2 4 6 8 10 12 0 0 2 4 6 8 10 12 y y Fig (12): Effect of rarefaction parameter on Fig (9): Effect of magnetic parameter on transient velocity for Pr=7 transient velocity for Pr=0.71 7
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Chemical and Process Engineering Research www.iiste.orgISSN 2224-7467 (Paper) ISSN 2225-0913 (Online)Vol 2, 2012 4.5 1.4 A=0 K=0.2, 4 A=0.5 K=0.4, 1.2 A=10 K=0.6, 3.5 K=0.8, A=15 1 3 Pr=0.71 2.5 0.8 2 u 0.6 Pr=0.71 1.5 Pr=7.0 1 0.4 0.5 Pr=7.0 0.2 0 0 -0.5 0 2 4 6 8 10 12 0 2 4 6 8 10 12 y y Fig (16): Temperature profiles against y Fig (13): Effect of suction parameter on transient velocity for Pr=0.71 and Pr=7. 1.4 7 Gc=1,Pr=7 1.2 Gc=5,Pr=7 6 Gc=10,Pr=7 Gc=15,Pr=7 1 1 5 2 4 0.8 u 3 3 0.6 4 2 5 0.4 6 1 0.2 0 0 2 4 6 8 10 12 y 0 Fig (14): Effect of mass Grashof number 0 2 4 6 8 10 12 on transient velocity for Pr=7 y Fig (17): Concentration profile against y 10 Gc=1,Pr=0.71 Gc=5,Pr=0.71 8 Gc=10,Pr=0.71 here Gc=15,Pr=0.71 Sc Kr A R 6 1. 0.22 0.1 5 2 2. 0.66 0.1 5 2 4 u 3. 0.98 0.1 5 2 4. 0.22 0.5 5 2 2 5. 0.22 1.0 5 2 0 6. 0.22 0.1 10 2 When ω=20, ωt = π/2, ε = .05, -2 Gr=Gc=5 0 2 4 6 8 10 12 y Fig (15): Effect of mass Grashof number on transient velocity for Pr=0.71References:[1]. Cramer, K. R. and Pai, S. I. Magneto fluid Dynamics for Engineering and Applied Physicists (Mc GrawHill, New York), (1973). 8
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Chemical and Process Engineering Research www.iiste.orgISSN 2224-7467 (Paper) ISSN 2225-0913 (Online)Vol 2, 2012[2] Muthucumaraswamy.R and Meenakshisundaram.S. Theoretical study of chemical reaction effects onvertical oscillating plate with variable temperature. Theoret. Appl. Mech., vol.33, No.3 (2006), pp.245-257.[3]. Sharma, P. K. Fluctuating thermal and mass diffusion on unsteady free convective flow past a verticalplate in slip-flow regime, Latin American Applied Research, 35, 313-319,(2005).[4] Anjalidevi, S.P. and R. Kandasamy. Effects of a chemical reaction heat and mass transfer on MHD flowpast a semi infinite plate. Z.Angew. Math. Mech.,(2000) 80:697-701.[5]. Chaudhary, R. C. and Jha, A. K.,Effects of chemical reactions on MHD micropolar fluid flow past avertical plate in slip-flow regime, Applied Mathematics and Mechanics, 29 (9), 1179-1194, (2008).[6] Muthucumaraswamy, R. and Chandrakala, P. Radiation heat and mass transfer effects on movingisothermal vertical plate in the presence of chemical reaction. J. of Applied Mechanics and Engineering, vol.11 NO.3, (2006), pp 639-646. [7]. Al-Odat, M. Q. and Al-Azab. Influence of chemical reaction on transient MHD free convection over amoving vertical plate, Emirates J. Engg. Res., 12 (3), 15-21. (2007).[8] M.A. El-Aziz, “Radiation effect on the flow and heat transfer over an unsteady stretching sheet”,International Communications in Heat and Mass Transfer, Vol. 36, pp 521-524, (2009).[9] P. Singh, N.S. Tomer and D. Sinha, “Numerical study of heat transfer over stretching surface in porousmedia with transverse magnetic field”, Proceeding of International Conference on Challenges and applicationof Mathematics in Sciences and Technology (2010), ISBN 023- 032-875-X, pp 422-430.[10] P. Singh, N.S. Tomer, S. Kumar and D. Sinha, “MHD oblique stagnation-point flow towards astretching sheet with heat transfer”, International Journal of Applied Mathematics and Mechanics, Vol. 6, no.13, pp 94-111, (2010).[11] P. Singh, A. Jangid, N.S. Tomer, S. Kumar and D. Sinha, “Effects of Thermal Radiation and MagneticField on Unsteady Stretching Permeable Sheet in Presence of Free Stream Velocity”, International Journal ofInformation and Mathematical Sciences, Vol.6, no.3, pp-63-69, (2010).[12] E. M. A. Elbashbeshy, D. M. Yassmin and A. A. Dalia, “Heat Transfer Over an Unsteady PorousStretching Surface Embedded in a Porous Medium with Variable Heat Flux in the Presence of Heat Sourceor Sink”, African Journal of Mathematics and Computer Science Research Vol. 3, no.5, pp 68-73, (2010).[13]. Kandasamy, R., Periasamy, K. and Sivagnana Prabhu, K.K. Chemical reaction, heat and mass transferon MHD flow over a vertical stretching surface with heat source and thermal stratification effects, Int. J.Heat and Mass Transfer, 48 (21-22), 4557-4561,(2005).[14]. Angirasa, D., Peterson, G. P. and Pop, I. Combined heat and mass transfer by natural convection withopposing buoyancy effects in a fluid saturated porous medium, Int. J. Heat Mass Trans., 40(12), 2755-2773,(1997).[15]. Ahmed, S. Effects of unsteady free convective MHD flow through a porous medium bounded by aninfinite vertical porous plate, Bull. Cal. Math. Soc., 99 (5), 511-522,(2007).[16]. Ahmed Sahin. Influence of chemical reaction on transient mhd free Convective flow over a verticalplate in slip-flow Regime. Emirates Journal for Engineering Research, 15 (1), 25-34 (2010) 9
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